On Reachable Families of the Loewner Differential Equation in Several Complex Variables

Iancu, Mihai

doi:10.1007/s11785-015-0461-z

Cited by 3 publications

(1 citation statement)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also mention the main contributions of Bracci [5] related to the existence of bounded support points for the family S 0 (B n ), n ≥ 2, and of Roth [31] concerning the ndimensional version of the well known Pontryagin maximum principle. Other recent contributions in the Loewner theory in C n may be found in [2,3,4,7,16,17,23,24,25,32].…”

Section: Introductionmentioning

confidence: 98%

Convergence results for families of univalent mappings on the unit ball in C^n

Hamada

Iancu

Kohr

2016

Ann. Acad. Sci. Fenn. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. Let S t A (B n ) be the family of normalized univalent mappings on the Euclidean unit ball B n in C n , which have generalized parametric representation with respect to time-dependent operators A ∈ A, where A is a family of measurable mappings from [0, ∞) into L(C n ) with some particular properties. Also, let, where A ∈ A and T 0 ≥ 0. In this paper we obtain certain convergence results for the families S t A (B n ) and) with respect to the Hausdorff metric ρ on H(B n ). These results may be seen as dominated convergence type theorems for time-dependent operators A ∈ A. In particular, we obtain related convergence results for the family S 0 A (B n ) (resp. for the family S A (B n )) of mappings with A-parametric representation on B n (resp. of spirallike mappings on B n with respect to A), in the case that A ∈ L(C n ) is a linear operator with k + (A) < 2m(A), where k + (A) is the Lyapunov index of A and m(A) = min z =1 ℜ A(z), z . We also obtain a convergence result for the Carathéodory family N A , where m(A) > 0. Finally, we obtain some sufficient conditions related to A ∈ A, which yield the equality S t A (B n ) = S 0 (B n ), for all t ≥ 0, where S 0 (B n ) is the family of normalized univalent mappings with usual parametric representation on B n . Certain consequences are also provided.

show abstract